Which of the following is the derivative of the function f(x)=sinx⋅(1−cosx) ? (1 point) Responses f′(x)=cosx⋅sinx f ' x = cos x · sin x f′(x)=−cosx⋅sinx f ' x = - cos x · sin x f′(x)=cosx+cosx⋅sinx f ' x = cos x + cos x · sin x f′(x)=cosx−cos2x−sin2x f ' x = cos x - cos 2 x - sin 2 x f′(x)=cosx−cos2x+sin2x
The derivative of the function f(x) = sinx⋅(1−cosx) is f'(x) = cosx⋅sinx + sinx⋅(−sinx) = cosx⋅sinx - sin^2(x).
Therefore, the correct answer is f'(x) = cosx⋅sinx - sin^2(x).
To find the derivative of the function f(x) = sin(x)⋅(1−cos(x)), we can use the product rule.
Let's break down the steps:
Step 1: Apply the product rule
The product rule states that if we have two functions, u(x) and v(x), the derivative of their product u(x)⋅v(x) is given by:
f'(x) = u'(x)⋅v(x) + u(x)⋅v'(x)
In this case, u(x) = sin(x) and v(x) = (1−cos(x)).
Step 2: Find the derivatives
The derivative of sin(x) is cos(x), and the derivative of (1−cos(x)) is sin(x).
Step 3: Plug into the formula
Plugging the derivatives into the product rule formula, we have:
f'(x) = (cos(x)⋅(1−cos(x))) + (sin(x)⋅sin(x))
Step 4: Simplify
f'(x) = cos(x)−cos^2(x) + sin^2(x)
Since sin^2(x) + cos^2(x) = 1 (a trigonometric identity), we can simplify further:
f'(x) = cos(x)−cos^2(x) + (1−cos^2(x))
= cos(x) − cos^2(x) + 1 − cos^2(x)
= cos(x) − 2cos^2(x) + 1
So, the derivative of f(x) = sin(x)⋅(1−cos(x)) is f'(x) = cos(x) − 2cos^2(x) + 1.
To find the derivative of the function f(x) = sin(x) * (1 - cos(x)), we can use the product rule and the chain rule.
First, let's simplify the function by multiplying and expanding the terms:
f(x) = sin(x) - sin(x) * cos(x)
Now, we can differentiate both terms separately.
For the first term, sin(x), the derivative is cos(x) (derivative of sin(x) is cos(x)).
For the second term, -sin(x) * cos(x), we need to use the product rule. The product rule states that if we have two functions, u(x) and v(x), then the derivative of their product is given by:
(u * v)' = u' * v + u * v'
Let's apply the product rule to our second term:
(u * v) = -sin(x) * cos(x)
u = -sin(x)
v = cos(x)
Using the product rule, the derivative of the second term is:
u' * v + u * v' = (-cos(x) * cos(x)) + (-sin(x) * -sin(x)) = -cos^2(x) + sin^2(x)
Putting it all together, the derivative of the function f(x) = sin(x) * (1 - cos(x)) is:
f'(x) = cos(x) - cos^2(x) + sin^2(x)
So, the correct answer is f'(x) = cos(x) - cos^2(x) + sin^2(x).